Isothermal Compressibility and Mean Square Fluctuation
a) Derive the relation
where kτ is the isothermal compressibility:
b) From (a), find the relation between kτ and the mean square fluctuation of N in the grand canonical ensemble. How does this fluctuation depend on the number of particles?
SOLUTION
a) Let us use the Jacobian transformation for thermodynamic variables:
Since the chemical potential μ is expressed in P, τ and does not depend on N, we can write
(1)
where s = S/N and v = V/N are reduced entropy and volume respectively. Using the equation for the Gibbs free energy of a single-component system, G = μN, we can write
where we also used V = (∂G/∂P)τ. But from (1),
So finally
b) By definition the average number of particles in the grand canonical ensemble is
where Ω = F – μN. Now, from (a),
where N is an average number of particles:
(2)
where we have used ⟨N⟩ = -∂Ω/∂μ. So
(3)
From (3)
and
Since V is proportional to ⟨N⟩,
The relative fluctuation is given by
